带小扰动的随机发展方程的不变测度

本文考虑带小扰动的随机发展方程,证明如何建立此方程的耦合解.作为应用,我们证明解的Feller连续性和不变测度的存在唯一性.还进一步建立了当扰动趋于零时,关于这族不变测度的大偏差原理.     

General random perturbations of hyperbolic and expanding transformations

Small random perturbations of a general form of diffeomorphisms having hyperbolic invariant sets and expanding maps are considered. The convergence of invariant measures of perturbations to the Sinaî-Bowen-Ruelle measure in the case of a hyperbolic attractor and to the smooth invariant measure in the expanding case are proved. The convergence of corresponding entropy characteristics and the approximation of the topological pressure by means of perturbations is considered as well.     

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.     

Alexander invariant for perturbations of Fredholm operators

The Alexander-type invariant for the bifurcation problems involving perturbations of linear Fredholm operators of nonnegative index is discussed and compared with another homotopy invariant (called the bifurcation index) detecting global bifurcations of solutions to multiparameter equations.     

Small perturbations of quasistochastic dynamical systems

Stability questions of invariant measures of quasistochastic dynamical systems are investigated under the action of small perturbations. Both purely random and deterministic perturbations, small in C, are considered. The connection between the stability properties of the invariant measures with respect to a given family of perturbations and the concept, introduced in the paper, of the intrinsic stability of these measures, determined by the dynamical system itself, is elucidated. The connection between the considered questions and the problems of the modeling of dynamical systems on computers is discussed.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 166–189, 1986.The author is greatful to Ya. G. Sinai for the formulation of the problem and for numerous useful discussions.     

Stochastic stability in some chaotic dynamical systems

For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the Perron-Frobenius-operator ofT that invariant measures depend continuously on 3 types of perturbations: 1) deterministic perturbations, 2) stochastic perturbations, 3) randomly occuring deterministic perturbations. The topology on the space of perturbed transformations is derived from a metric on the space of Perron-Frobenius-operators.With 1 Figure     

Energy criterion of global existence for energy-critical Schrödinger equations with subcritical perturbations

We present a variational approach to study the energy-critical Schrödinger equations with subcritical perturbations. Through analysing the Hamiltonian property we establish two types of invariant evolution flows, and derive a new sharp energy criterion for blowup of solutions for the equation. Furthermore, we answer the question: how small are the initial data such that the solutions of this equation are bounded in H ¹(R ^N )?     

A Method of Defining Central and Gibbs Measures and the Ergodic Method

Doklady Mathematics - We formulate a general statement of the problem of defining invariant measures with certain properties and suggest an ergodic method of perturbations for describing such...     

Wasserstein convergence of invariant measures for fractional stochastic reaction–diffusion equations on unbounded domains

This paper is concerned with the convergence of invariant measures in the Wasserstein sense for fractional stochastic reaction–diffusion equations defined on unbounded domains as the noise intensity approaches zero. Based on uniform estimates of solutions, we prove the family of invariant measures of the stochastic equations converges to the invariant measure of the corresponding deterministic equations in terms of the Wasserstein metric. We also provide the rate of such convergence.     

On the asymptotic distribution of eigenvalues of banded matrices

We consider the abstract measures, known as thedensity- of- states measures, associated with the asymptotic distribution of eigenvalues of infinite banded Hermitian matrices. Two widely used definitions of these measures are shown to be equivalent, even in the unbounded case, and we prove that the density of states is invariant under certain, possibly unbounded, perturbations. Also considered are measures associated with the asymptotic distribution of eigenvalues of rescaled unbounded matrices. These measures are associated with the so-called contracted spectrum when the matrices are tridiagonal. Finally, we produce several examples clarifying the nature of the density of states.Communicated by Paul Nevai.     

Stochastic predator–prey model with Allee effect on prey

We study a predator–prey model with the Allee effect on prey and whose dynamics is described by a system of stochastic differential equations assuming that environmental randomness is represented by noise terms affecting each population. More specifically, we consider a term that expresses the variability of the growth rate of both species due to external, unpredictable events. We assume that the intensities of these perturbations are proportional to the population size of each species. With this approach, we prove that the solutions of the system have sample pathwise uniqueness and bounded moments. Moreover, using an Euler–Maruyama-type numerical method we obtain approximated solutions of the system with different intensities for the random noise and parameters of the model. In the presence of a weak Allee effect, we show that long-term survival of both populations can occur. On the other hand, when a strong Allee effect is considered, we show that the random perturbations may induce the non-trivial attracting-type invariant objects to disappear, leading to the extinction of both species. Furthermore, we also find the Maximum Likelihood estimators for the parameters involved in the model.     

A global bifurcation index for set-valued perturbations of Fredholm operators

In the paper the homotopy invariant detecting global bifurcations of solutions to multi-parameter equations involving general set-valued perturbations of linear Fredholm operators of non-negative index is introduced. Some applications to the existence problems for differential inclusions are provided.     

Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise

In this paper, we consider the ergodicity of invariant measures for the stochastic Ginzburg-Landau equation with degenerate random forcing. First, we show the existence and pathwise uniqueness of strong solutions with H¹-initial data, and then the existence of an invariant measure for the Feller semigroup by the Krylov-Bogoliubov method. Then in the case of degenerate additive noise, using the notion of asymptotically strong Feller property, we prove the uniqueness of invariant measures for the transition semigroup.     

Stability for a random evolution equation with Gaussian perturbation

In this paper we consider a random evolution equation which is perturbed by Gaussian type noise, and show how to construct its coupled solutions. On the basis of the coupling results, we discuss the asymptotic flatness and the stability in total variation norm for the solutions of the equation. In addition, we also prove the Feller continuity, and the existence and uniqueness of invariant measures of the solutions.     

Floquet torus for codimension-<Emphasis Type="Italic">s</Emphasis> Hopf bifurcation

Here we show the existence of Floquet invariant torus for a codimension s Hopf bifurcation. As a corollary is obtained the existence of Floquet invariant torus for generic perturbations of product of quadratic maps.     

Synthesis of nonlinear control systems invariant under perturbations

For nonlinear control systems with perturbations, we consider the problem of synthesis of perturbation-invariant characteristics (invariant functions) with the use of feedbacks. The existence of invariant functions is related to a decomposition of the control system for which the quotient system is independent of perturbations. We present conditions for the existence of such quotient systems, which are certain systems of partial differential equations. The synthesizing controls are found from these equations.     

Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly (Ann. Math. 164:993–1032, 2006), we prove the uniqueness of invariant measures for the corresponding transition semigroup.     

Large deviations for invariant measures of SPDEs with two reflecting walls

In this article, we establish a large deviation principle for invariant measures of solutions of stochastic partial differential equations with two reflecting walls driven by a space–time white noise.     

Persistence of Lower Dimensional Tori for a Class of Nearly Integrable Reversible Systems

In this paper we consider the persistence of lower dimensional invariant tori for a class of reversible systems, and prove that if the average of the linear terms has full rank, then the invariant torus with Diophantine frequencies persists under small perturbations.     

ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS

It is well known that the analysis of the large-time asymptotics of Fokker-Planck type equations by the entropy method is closely related to proving the validity of convex Sobolev inequalities. Here we highlight this connection from an applied PDE point of view.

In our unified presentation of the theory we present new results to the following topics: an elementary derivation of Bakry-Emery type conditions, results concerning perturbations of invariant measures with general admissible entropies, sharpness of convex Sobolev inequalities, applications to non-symmetric linear and certain non-linear Fokker-Planck type equations (Desai-Zwanzig model, drift-diffusion-Poisson model).